Question: What is the period of $y=-5\cos\left(\dfrac{\pi}{8} x\right)+3$ ? Give an exact value. units
Solution: Period in sinusoids of the form $y=a\cos(bx+c)+d$ Graphically, the period of a sinusoidal function is the horizontal distance between the ends of a single cycle of its graph. The period of a sinusoid of the form $y={a}\cos( bx + c) + {d}$ is equal to $\dfrac{2\pi}{| b|}$. [How can we justify this given our graphical understanding of period?] Finding the period The period of $y=-5\cos\left({\dfrac{\pi}{8}} x\right)+3$ is: $\begin{aligned} \text{period}&=\dfrac{2\pi}{|{b}|}\\\\ &=\dfrac{2\pi}{\left|{\dfrac{\pi}8}\right|} \\\\\\\\\\ &=2\pi\cdot \dfrac{8}{\pi} \\\\ &=16 \end{aligned}$ The answer The period of $y=-5\cos\left(\dfrac{\pi}{8} x\right)+3$ is $16$ units.